Find the domain of the function.
​P(t)equalsStartFraction StartRoot t minus 4 EndRoot Over 2 t minus 16 EndFraction
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Part 1
The domain is 
  
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To find the domain of the function \( P(t) = \frac{\sqrt{t - 4}}{2t - 16} \), we need to consider two things:

1. **Square Root Condition**: The expression inside the square root, \( t - 4 \), must be non-negative because we cannot take the square root of a negative number in the real number system. Thus, we need:

   \[
   t - 4 \geq 0 \implies t \geq 4
   \]

2. **Denominator Condition**: The denominator \( 2t - 16 \) must not be zero because division by zero is undefined. So, we solve:

   \[
   2t - 16 \neq 0 \implies t \neq 8
   \]

### Domain:
- From the square root condition, \( t \geq 4 \).
- From the denominator condition, \( t \neq 8 \).

Thus, the domain of \( P(t) \) is all \( t \geq 4 \) except \( t = 8 \).

In interval notation, the domain is:

\[
[4, 8) \cup (8, \infty)
\]

Would you like further details or have any other questions?

Here are 5 related questions for deeper understanding:
1. How would the domain change if the square root term were \( t - 5 \) instead of \( t - 4 \)?
2. What happens to the domain if the denominator is \( 2t - 12 \)?
3. What is the range of this function \( P(t) \)?
4. How would the domain be affected if the square root were in the denominator instead?
5. Can you explain how the domain would change if there were an absolute value inside the square root?

**Tip**: Always check both the square root and the denominator when determining the domain of a function involving these components.

Find the domain of the function. P(t) = sqrt(t - 4) / (2t - 16)

Learn how to find the domain of the function P(t) = sqrt(t - 4) / (2t - 16) with a detailed solution. This problem covers key mathematical concepts, such as square root conditions and rational functions. Suitable for students in Grades 10-12.

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